In my spare time lately, I've been toying with ways to coherently deal with infinite quantities in ways that follow most of the same rules as finite quantities, and to assign a meaningful interpretation to such manipulations. I've come up with something that I think makes sense, and I couldn't find anything quite like it on Wikipedia, so I send it to the forum to check my work. Warning: lacks rigor.

The quantity of any finite number of points in a space is simply the cardinality of the set of those points.

Let R1 be the infinite quantity of points in the half-open interval [0,1).

In general, R1 is to be assumed to obey all the algebraic laws normally valid for positive real numbers. Conclusions about the size(quantity of points) of more complex shapes are drawn from the above definition, and the principle that the size of any set measured in this way will be translation-invariant, rotation-invariant, and reflection-invariant within the relevant space. Shapes that cannot be directly constructed will be taken to be the limit of appropriate approximations to those shapes (appropriate in the sense that the troll proof of pi=4 by using rectangular approximations is an inappropriate approximation. The first-order derivative at the boundary must also converge in the limit)

R12 is then the half-open square of [0,1)x[0,1). The size of the closed interval [0,1] is R1+1, and the size of the closed unit square is R12+2R1+1. The size of [0,n) is n*R1, the size of an n x m half-open rectangle is n*m*R12, and so on for higher dimensional "rectangles".

Arbitrary polygons can be decomposed into triangles, and arbitrary triangles can be formed by cutting across the diagonal of a rectangle, the size of which is defined above. But care must be taken to ensure that the two halves have exactly the same number of points, including the points along the perimeter. An R12 term, when taken to be describing a triangle, is satisfied by a set containing the entire interior of the triangle, exactly half of the perimeter points, with 2 discrete endpoints on that section of perimeter. But any other construction that can be shown to be the same size (by translation/rotation/reflection of subsets) is valid as well. The most convenient such construction is a set containing the entire interior of the triangle *less* 1 interior point, and a half-open interval from the midpoint of each edge to the corner. One extra endpoint is given so that each edge has its own half-open interval, but this is accounted for by removing one point from the interior.

Given this construction, we can draw conclusions about arbitrary arrangements of polygons. Any two triangles joined at an edge will each contribute half the length of that edge, overlapping at the midpoint. This extra point can be shifted to fill the missing interior point of one of the triangles. With the shared edge completed, and one interior hole filled, the resulting quadrilateral is constructed in much the same way as the triangles were. There is one hole in the interior, and half of each edge, with no corners filled. The same logic extends to arbitrarily large connections of polygons. Since every exterior edge is missing precisely a half-interval from corner to midpoint, the entire boundary is completed by adding an R11 term, leaving only a single point to fill the interior hole and complete the bounded shape.

But something interesting happens when we consider polygons arranged to form the boundary of a polyhedron. Each face(F) has one missing interior point. Each edge(E), at which two polygons meet, will have an extra point where the half-open intervals overlap. And by construction each vertex(V) is left initially unfilled. So in order to complete the polyhedron, we must add a number of discrete points equal to F-E+V, which is exactly the classical definition of the Euler characteristic.

Looking at the shapes we've tried so far, the number of discrete points always matches the Euler characteristic. For a discrete set of points, it's the cardinality. For a closed interval, it's 1 (the missing point from the half-open interval of the R1 term). For a polygonal perimeter, it's 0. For any filled polygon, it's 1. And taking simple smooth shapes like the circle, disc, and sphere to be the limits of their polygonal/polyhedral approximations, the correct values hold. After much difficult visualization of how to apply similar generic construction rules to 3d shapes, I determined that a pure R13 term interpreted as a polyhedral solid would have an extra point in its interior, which is in keeping with the extension of the classical Euler characteristic to higher dimensions. Proving it rigorously proves thus far out of my grasp, but it seems to be a reasonable conjecture and operational assumption that the finite term of the size of a given set will be equal to the Euler characteristic of that set.

It is also fairly obvious from the examples thus far explored that the coefficient of the R1n term of an n-dimensional object is equal to its measure in the traditional sense. So that just leaves the intermediate terms.

It's easy to show that the R1n-1 coefficient is half of the traditional measure of the boundary of the shape (perimeter in 2d, surface area in 3d, etc). This can be seen from the exercise of splitting the half-open n-cube. The boundary between the two halves is part of the interior of the original shape, so the two halves must each get half of that boundary, which is consistent with the fact that the half-open n-cube already included half of its boundary.

But R1m where 0<m<n-1 is more difficult to figure out. The first time such a term even exists is in the case of finding the coefficient of R11 for a 3d solid. Although it is extremely difficult(maybe impossible) to deduce directly from slicing up the half-open cube, the result can be reasoned out by assuming that *some* construction rule must exist that will be consistent with the n-cube itself, and that has the property that pasting two such shapes together at a shared face will yield a larger solid that fits the same construction rule.

The rule that works is that a proportion of each edge is included, and that proportion is equal to the proportion of the circle subtended by the dihedral angle around that edge. So, for the cube, the dihedral angle is tau/4 (tau=2pi), so 1/4 of each edge is included. These edges can be cut-and-pasted to have 3 full edges, 1 along each axis, as there would be for the direct construction of the cube based on the product of half-open intervals. And, when constructing larger polyhedra using tetrahedra, much as one could use triangles to construct polygons, the only restriction on the number of tetrahedra sharing a given edge in the interior of the resulting solid is that their dihedral angles must add up to a complete revolution, or else the edge would still be exposed to the exterior of the solid. When this condition is met, the fractions of the edge contributed by each tetrahedron will by definition sum to 1, meaning the entire edge is now included, since it's just another part of the interior.

The only edges that aren't filled are those on the exterior. And then, in addition to the proportion contributed by the R13 term, exactly 1/2 of the total edge length will be contributed by the R12 term (the R12 term will be half the surface area, and any R12 term of a polyhedral surface area will include all the edge length. This particular identity I've shown to myself will hold in a general construction, where it doesn't matter "which half" of the surface area comes from R13 or R12, but I won't go into it here). When taking polyhedral approximations to a smooth surface, the dihedral angle at the surface approaches tau/2, so the R13 term will contribute half of that edge, and the R12 term will contribute the other half, meaning the entire edge is included. So taking the limit of approximations to the sphere ends up needing no additional term for edge length, which is of course expected since a sphere has no edges. So far the system is consistent with expected results.

This approach to working out what proportion of the exterior edges are included in the R13 term can be extended to higher dimensions. The notion of the dihedral angle can be extended to be the proportion of the k-sphere subtended by the interior of the solid, where k is whatever dimensionality is necessary for the angle. In general, the proportion of the m-dimensional boundary of an n-dimensional set that is included by the R1n term is the proportion of the (n-m)-sphere subtended by the (n-m)-dimensional "dihedral angle" around that boundary, or the integral of said angle for all points on that boundary, if it is not constant. This even extends to the R1n-1 case. Only the R10 case doesn't fit easily, but that's fine because it doesn't make sense to take a proportion of a single point anyway, and we already have the Euler characteristic as a general way of finding that term.

Now that there's a way to determine what proportion of each type of boundary is included by the higher dimensional terms, they can simply be subtracted off from the total value to find the coefficient of the relevant power of R1. Meaning there is now a general method of calculating the size of all manner of sets.

It also seems to work reasonably for fractals. When the value of a coefficient diverges in the limit, define the coefficient itself to be an infinite quantity obtained by taking the approximation where edge lengths are R1-1, or areas are R1-2, or whatever's appropriate for the construction. Based on the principle that n*R1 is supposed to be the number of points in the half open interval of length n, 1/R1 is the "length" that results in a single discrete point, making a perfect approximation to the desired shape. Likewise for reducing areas to discrete points, and so on. This hand-wavy technique is apparently coherent enough to give the correct fractal dimension of the Koch curve (the only example I worked out), so it seems to work.

So, is this a well-known thing that I simply didn't know what term to search for, or am I onto something new here, or is it completely broken in some way that I haven't seen?

EDIT:Actually, I'm not sure what I was thinking when I said that there ends up being no length term on the sphere, and that that is an expected result because the sphere has no edges. The sphere has no vertices either, but it still has a finite term. And upon actually trying to construct various smooth solids as limits of their approximations, I do see length terms appearing that don't correspond to actual edges of the resulting solid. But they do seem to be consistent regardless of the specifics of the sequence of approximations, and they seem to be meaningful 1-dimensional characteristics of the shape in question. The height of a cylinder, the length along the surface from the vertex to the base of a cone, half the circumference of a sphere, etc. Also, the general method of finding such terms by subtracting out the amount included by the higher dimensional terms does in fact work for the 0-dimensional case. This seems to define a generalization of the Euler characteristic, where the regular Euler characteristic is the 0-dimensional term, and can be computed by the same generalized method as the higher-dimensional terms.

tl;drI define a number R1 to be the quantity of points in a half-open unit interval, assume R1+1 != R1 and so on for higher powers of R1, and use various shenanigans to build a method of computing the exact quantity of points in arbitrary shapes in Euclidean space given this notion of infinite quantities of points.

arbiteroftruth wrote:The quantity of any finite number of points in a spaaace is simply the cardinality of the set of those points.

Ok, you made a definition. For a finite set, the set's "quantity" is the set's cardinality.

arbiteroftruth wrote:Let R1 be the infinite quantity of points in the half-open interval [0,1).

Oh but I don't know what that means. You did not define quantity for an infinite set.

arbiteroftruth wrote:In general, R1 is to be assumed to obey all the algebraic laws normally valid for positive real numbers.

If by quantity you mean cardinality, that isn't going to be true in general. You can't sensibly subtract, for one thing; and you can add 1 to an infinite cardinality without changing the cardinality.

But if by "quantity" you mean something OTHER than cardinality, you haven't said what it is. So I stopped reading here.

I did skim forward a bit and it seems that you're adding 1 to your infinite quantities. You won't be able to make this work unless you define what you mean by quantity. If you mean cardinality, the laws of infinite cardinals are already well known and they don't work the same as real numbers.

[meta -- is this site slower than heck lately? Something at the server end?]

Allow me to rephrase. I denote the quantity of points in [0,1) to be this thing called R1, and I define R1 to follow the same algebraic rules as positive real numbers. I then assume for the sake of moving forward that this does not create any contradictions.

The entire definition is just that it is a notion of the "size" of the set that will follow regular algebraic rules, unlike cardinality.

arbiteroftruth wrote:Allow me to rephrase. I denote the quantity of points in [0,1) to be this thing called R1, and I define R1 to follow the same algebraic rules as positive real numbers. I then assume for the sake of moving forward that this does not create any contradictions.

The entire definition is just that it is a notion of the "size" of the set that will follow regular algebraic rules, unlike cardinality.

Yes but you can't just make a definition, you have to show that it makes sense. Since a lot of smart people have looked at the same problem and all agreed that cardinality is the best we can do here, the burden is on you to say what you mean.

There are other measures of set size ... measure, for one. In measure theory the size of the interval [0,1] is 1; and you can add sizes: the size of the union of [0,1] and [3,7] is 1 + 4 = 5.

But you can't just say that R1 is the "quantity" of points in [0,1] or [0.1) without telling us what you mean by quantity. The rules of cardinal arithmetic are already well known and they do NOT follow the usual laws of arithmetic on the real numbers.

So you have to at least provide some indication that you think there's something called "quantity" that measures the size of infinite sets better than cardinality does. It's true that IF there were such a thing, and X was a quantity, then we'd have X < X + 1. But we don't have any such convenient measure for infinite sets.

arbiteroftruth wrote:Allow me to rephrase. I denote the quantity of points in [0,1) to be this thing called R1, and I define R1 to follow the same algebraic rules as positive real numbers. I then assume for the sake of moving forward that this does not create any contradictions.

The entire definition is just that it is a notion of the "size" of the set that will follow regular algebraic rules, unlike cardinality.

Yes but you can't just make a definition, you have to show that it makes sense. Since a lot of smart people have looked at the same problem and all agreed that cardinality is the best we can do here, the burden is on you to say what you mean.

Seeing how much sense that definition makes is the entire point of the thread, and the content of the 90% of the original post that you skipped over.

I'm not trying to publish a math paper here. I'm just tossing out an idea that may or may not make sense upon closer scrutiny, and asking for feedback.

fishfry wrote:There are other measures of set size ... measure, for one. In measure theory the size of the interval [0,1] is 1; and you can add sizes: the size of the union of [0,1] and [3,7] is 1 + 4 = 5.

Which is exactly the reason for the thread title. The notion of 'quantity' is meant to resemble measure, but with the main alteration that the 'quantities' of [0,1] and [0,1) differ by 1, and other related consequences of this. The quantity of points in a given set in (at least Euclidean) space are defined to be translation invariant, rotation invariant, and reflection invariant. Within that context, disjoint set union should result in an addition of the quantities of the individual sets(if they can be quantified in the first place, just as there can be unmeasurable sets, there might also be unquantifiable sets). Whenever countable additivity converges, the quantity should be additive. And the quantity of a smooth curve is taken to be the limit of the quantities of polygonal/polyhedral approximations to that curve, given that the first derivative of the boundary of the curve is also being reached in the limit of the approximations (so no rectangular approximations to a circle to 'prove' that pi=4).

The infinite number R1 is necessary in order to have something that acts like measure while still distinguishing [0,1] from [0,1). If individual points have non-zero quantity even relative to a continuous line segment, but the rules of translation invariance and additivity allow me to cram infinitely many discrete points inside a given line segment, then the quantity of the line segment must be larger than any integer number of points. But because it otherwise is meant to behave like measure, it should follow the same basic algebraic rules as positive real numbers. Hence it is appropriate to introduce R1 as a notation to specify the infinite quantity of points in [0,1), so that we can algebraically manipulate it without equating it with a finite number.

EDIT: Perhaps a useful way to describe what I'm thinking of is that I'm essentially trying to do measure theory with hyperreal values of measure rather than real values, and a discrete point is always given a value of 1, even in the context of a continuum. As a consequence, any value we assign to a half-open interval like [0,1) must be an infinite hyperreal. I arbitrarily select an infinite hyperreal denoted R1 for this purpose, and the hyperreal measure of other sets is defined relative to that initial arbitrary selection.

I admit that I haven't read through your whole post yet (I'm distracting myself from a midterm at the moment), but I see an immediate issue. What's the quantity of [0,n]? From your definitions, it seems that we can either say that the quantity of [0,n) is n*R1, so the quantity of [0,n] is n*R1+1, or we can say that the quantity of [0,1] is R1+1, so the quantity of [0,n] is n*(R1+1)=n*R1+n. This is definitely going to lead to contradictions. (Well, the obvious fix is to say that the quantity of [0,1) is the same as the quantity of [0,1], but then you probably just get measure theory.)

MostlyHarmless wrote:or we can say that the quantity of [0,1] is R1+1, so the quantity of [0,n] is n*(R1+1)=n*R1+n.

No, because (according to the definition given above) you are counting {1,2,3,...[n]-1} and maybe [n] twice.I feel like there is a reason why this definition doesn't work, because it's said that measure is the best we can do (in real numbers? I'm not sure). I'm not even sure if your thing defines a field -- I think it doesn't, only defines a ring.

For the sake of sanity, I am going to denote R1 as x, as I don't see where a R2 would come in.

I cannot find any counterexamples in the real line.I get the quantity of a cube [0,1]3 as x3+3x2+3x+1. Half the surface area, half the edges, and some random point.I also get x3 to be the half-openish cube [0,1)3; not sure what you mean by "having an extra point in its interior".

(reads your thing more carefully) Wait...

(0,1) is x-1; (0,1)3 is an open cube and has quantity x3-3x2+3x-1; the boundary of this cube, then, is 6x2+2 (actual surface area * x2 + euler characteristic), which seems to line up with what you are saying. Even though I constructed it in a completely different way than you did.

I'm waiting anxiously to see where it falls, so I can be pleasantly surprised when it actually works.

Also, to the nitpickers above; I think ℤω with reverse lexographic ordering works as what quantity maps to, with ℤω being the set of all sequences over the integers that end in a sequence of zeroes.

I think this is an interesting idea. However, without being an expert in measure theory, I can immediately see a (potential) problem. Now, this may not be critical, but nevertheless: let the quantity of a set [imath]S[/imath] be [imath]q(S)[/imath].

This may be "bad", in the sense that additivity is a common requirement for measures and their generalizations. However, what you really want to achieve is find a true contradiction in [imath]R_1[/imath]. I believe this may be achieved, somehow, by "removing points" such as in my example above, by dividing the open interval (0,1) indefinitely and taking a sequence of quantities over these.

Also, I think an interesting question is: let Q be all rational numbers in (0,1). What is the quantity of (0,1)\Q?

Elmach wrote:I feel like there is a reason why this definition doesn't work, because it's said that measure is the best we can do (in real numbers? I'm not sure). I'm not even sure if your thing defines a field -- I think it doesn't, only defines a ring.

If quantity can be consistently represented as hyperreal numbers, then it should be a field.

But there are definitely some things measure theory can do that "quantity theory" can't, at least not without adding a few new concepts. For example, in measure theory, you can remove a countably infinite set of points from [0,1) and still give the resulting set a measure of 1. In quantity theory, the quantity of points removed, and thus the quantity of the resulting set, does not converge unless you identify the quantity of points removed with a specific infinite hyperreal. So, even if quantity theory is consistent, the ability to distinguish [0,1) from [0,1] does come with a tradeoff.

Elmach wrote:For the sake of sanity, I am going to denote R1 as x, as I don't see where a R2 would come in.

Mostly I was using R1 to denote specifically the infinity of the unit interval on the real number line, with the option of introducing other notation for other infinite quantities that might be difficult to compare with each other. Like the example of removing a countably infinite number of discrete points out of a continuous interval. I might conceivably be able to identify this with another infinite hyperreal that can't be expressed in terms of R1. It might also be useful to use something like N for the quantity of natural numbers, to help extend quantity to unbounded sets, but that opens a whole new can of worms. Even if I do figure out how to make that work, that's a long way off. x is fine for now.

Elmach wrote:I cannot find any counterexamples in the real line.I get the quantity of a cube [0,1]3 as x3+3x2+3x+1. Half the surface area, half the edges, and some random point.I also get x3 to be the half-openish cube [0,1)3; not sure what you mean by "having an extra point in its interior".

The "extra point in the interior" results from rearranging boundary points included in x3 in order to get a more symmetrical set that would be more suitable for constructing arbitrary shapes. x3 includes the entire interior of the cube, 3 faces, 3 edges, and only 1 vertex. You can take each of the 3 edges (one along each axis), and split it into 4 half-open intervals, each covering 1/4 of one of the 4 edges parallel to the original. That is, the cube includes the interval [<0,0,0>,<1,0,0>), but there are 4 edges that extend along the x dimension. So that interval can be split and shared among all 4 of those edges to get a more symmetrical set.

That gives a total of 12 quarter-edges, but still only 10 endpoints between them, since three of them share an endpoint at <0,0,0>. Those three can be split off to each be a half-open interval with its own unshared enpoint, but to do so requires that we borrow 2 points from somewhere else in the cube.

The cube includes 3 faces out of 6. Each face can be split in 2 halves, one for itself and one for the opposite face. But the pure interior of a face doesn't split cleanly in two. Cutting through the face produces an open interval at the cut, which itself must be split in half, but an open interval is of the form r*x-1, which can't be split in half, because that would require having half of a discrete point. But if we first remove a single point from the open interval formed by the cut, then we have something of the form r*x-2, which can be split in half. So in order to split of the faces for more symmetry, we have to remove one point from each of the 3 faces that were included in x3. 2 of these points can be used as the 2 points we needed to borrow in the paragraph above. This leaves us with 1 orphan point that we have to put somewhere in the cube. So, just make it an extra point in the interior.

Now, x3 is rearranged to be completely symmetrical. It contains the entire interior, half of each face, a quarter of each edge, none of the vertices, and 1 spare point in the interior. This is useful because it can be generalized to more complicated polyhedra that are difficult to construct from rectangular prisms. A generic r*x3 term, interpreted as a polyhedron, will contain the entire interior, half of each face, none of the vertices, one extra point in the interior, and a proportion of each edge equal to the proportion of the full circle subtended by the dihedral angle around that edge. In the case of the cube, each dihedral angle is a quarter-turn, so one quarter of each edge is included.

This is quite interesting, and I don't see anything immediately and obviously wrong with it.

I do have an observation, though:

[0,1) has quantity R1Moving the point at 1/2 to 1, you end up with [0,1/2)∪(1/2,1], which still (presumably) has quantity R1.Moving the point at 1/3 to 1/2, you end up with [0,1/3)∪(1/3,1], which still (presumably) has quantity R1.Moving the point at 1/4 to 1/3, you end up with [0,1/3)∪(1/3,1], which still (presumably) has quantity R1....In general, moving the point at 1/n to 1/(n-1), you end up with [0,1/n)∪(1/n,1], which still (presumably) has quantity R1.

Repeating countably many times, you end up with [0,1], but this has quantity R1+1.

What this is actually doing is constructing a bijection between [0,1) and [0,1]; explicitly, it is this:f: [0,1) -> [0,1]f (1/n) = 1/(n-1) for all integers n > 1f (x) = x for all other x

Presumably, this means that, if this notion of quantity is to make any sense, then you can't reason about 'infinite' processes in this way.

Also, I'm not entirely sure I understand the talk about fractals and R1-n. Could you explain how you calculate the quantity of, for example, the Cantor set?

benneh wrote:This is quite interesting, and I don't see anything immediately and obviously wrong with it.

I do have an observation, though:

[0,1) has quantity R1Moving the point at 1/2 to 1, you end up with [0,1/2)∪(1/2,1], which still (presumably) has quantity R1.Moving the point at 1/3 to 1/2, you end up with [0,1/3)∪(1/3,1], which still (presumably) has quantity R1.Moving the point at 1/4 to 1/3, you end up with [0,1/3)∪(1/3,1], which still (presumably) has quantity R1....In general, moving the point at 1/n to 1/(n-1), you end up with [0,1/n)∪(1/n,1], which still (presumably) has quantity R1.

Repeating countably many times, you end up with [0,1], but this has quantity R1+1.

What this is actually doing is constructing a bijection between [0,1) and [0,1]; explicitly, it is this:f: [0,1) -> [0,1]f (1/n) = 1/(n-1) for all integers n > 1f (x) = x for all other x

Presumably, this means that, if this notion of quantity is to make any sense, then you can't reason about 'infinite' processes in this way.

Also, I'm not entirely sure I understand the talk about fractals and R1-n. Could you explain how you calculate the quantity of, for example, the Cantor set?

Regarding the process of moving a single missing point closer and closer to 0, informally, I would argure that the quantity applies to real numbers within the interval, but it is treated as being imbedded in the hyperreal number line. So that as the missing point becomes smaller, it doesn't converge to 0, but to some infinitismal non-zero, and more importantly non-real, number. Meaning that in the limit, the point in question leaves the set of real numbers, thus breaking the invariance of quantity in the infinite process. Whereas using something like a process of polynomial approximations to the circle, one could say that the half-open interval representing a side extending from <1,0> converges to the discrete point <1,0>, and the adjacent side converges to a discrete point with non-real coordinates, nevertheless the total quantity of real-coordinate points on the circle converges to R1 times the conventional circumference.

The Cantor set turns out to have quantity R1ln(2)/ln(3). Construct it as the limit of sequences of closed intervals according to the following pattern.Iteration 1: [0,1]Iteration 2: [0,1/3]U[2/3,1]Iteration 3: [0,1/9]U[2/9,1/3]U[4/9,2/3]U[8/9,1]And so on.

In the limit, the smallest that a given half-open interval could become is a single point This occurs when the half-open interval has a hyperreal lengh of R1-1. The quantity of a half-open interval of length l is l*R1, so a half-open interval of quantity 1 has a length of R1-1.

For a given length l of the constituent closed intervals, the quantity of the approximation to the Cantor set is 2-ln(l)/ln(3)*(l*R1+1). In the limit, l=R1-1, and substituting this into the formula gives a quantity of 2*R1ln(2)/ln(3), after some manipulations.

But consider the interval containing 0. In the limit, the hyperreal half-open interval is [0,R1-1), which contains exactly 1 real number as desired. But the approximation always had closed intervals, and the closure of this interval is [0,R1-1], which still contains exactly 1 real number. Just as in your observation of the missing point that shrinks toward 0, the second endpoint of the interval leaves the set of real numbers and thus no longer contributes to the quantity. Similarly, the interval containing 1 becomes [1-R1-1,1)U[1], and this time it is the half-open intterval that vanishes, containing 0 real numbers. Every closed interval thus converges to a single point, rather than 2 points(one from the half-open interval and one from the closure). So the quantity is only half what we computed algebraically.

Quantities are assigned to sets of real numbers embedded in a hyperreal space. So if you define a process where, in the limit, some of your points take on hyperreal values, those points will no longer contribute to the quantity, and quantity invariance of the process can be broken. In the case of your observation, the interpretation is that the 'gap' in the interval takes on a hyperreal infinitesimal value in the limit.

EDIT: Here's a justification of that interpretation. Your process consistently has a form of [0,r)U(r,1]. An interval that converges to a discrete point in the limit of a process must do so by converging to an interval of length R1-1, to maintain consistency with the formulas regarding length and quantity of intervals. So if your process is to retain the form of two half-open intervals with a shared open boundary, and the left interval is to take on a quantity of 1 at the point 0, then the set must be [0,R1-1)U(R1-1,1], in which the 'gap' has left the set of real numbers.

Maybe you could say a little about what you mean by hyperreals. I've heard of the hyperreals in nonstandard analysis, in which each real number has a little cloud of infinitesimals around it. I've also heard of Conway's Surreal numbers, which can express various transfinite quantities. But I don't know too much about either of those systems.

Can you say exactly what hyperreals you are using and what are their properties? And why you think they may be the key to a new, improved "better than cardinality" counting technique for infinite sets?

If you start with a diamond, ie the convex hull on the points (1, 0), (0, 1), (-1,0) and (0, -1), and then fill it with sets of the form [a,b)x[c,d) then we can fill up the whole set, aside from 3 of its sides. So it has "quantity" 2R12 + 3sqrt(2)(R1) + 3. This is different from the result when we align the shape with the axes. So quantity isn't rotation invariant?

jestingrabbit wrote:If you start with a diamond, ie the convex hull on the points (1, 0), (0, 1), (-1,0) and (0, -1), and then fill it with sets of the form [a,b)x[c,d) then we can fill up the whole set, aside from 3 of its sides. So it has "quantity" 2R12 + 3sqrt(2)(R1) + 3. This is different from the result when we align the shape with the axes. So quantity isn't rotation invariant?

Err... what you said gives me 2x2 + 3sqrt(2)x + 1.

But that's just a quibble, because when rotated, I get 2x2 + 2sqrt(2)x + 1.

At this point, quantity is undefined for unbounded sets like the natural numbers. I've put some effort into trying to make that work, but it brings it's own challenges. For example, if N denotes the quantity of natural numbers, then by using an argument from translation invariance, N+1=N. So it can't be a standard hyperreal number, and things get much more difficult.

Elmach:

Given the construction method I already posted for approximating the Cantor set, at iteration n of the approximation, the quantity is 2n*(3-nx+1). Taking the limit as n approaches an unspecified infinity simply gives 2^(unspecified infinity), which is not helpful. In order to have a meaningful answer, we have to identify n as taking on a specific infinite hyperreal value in the limit.

The construction consists of line segments of length 3-n. In the limit, we want these line segments to become discrete points of the cantor set, meaning each line segment should have a quantity of 1. The quantity of a line is x*length, so a line segment with quantity 1 has length x-1. So to take the construction to the point at which this occurs, we set 3-n=x-1. Solving for n gives n=-ln(x-1)/ln(3)=ln(x)/ln(3).

Substituting this value for n into the formula for quantity at a given iteration gives 2ln(x)/ln(3)(3-ln(x)/ln(3)*x+1). The parenthetical term reduces to 2, and using the identity alog(b)=blog(a) lets us get x out of the exponent on the left term. This gives 2*xln(2)/ln(3).

That factor of 2 goes away upon analysis of which points in the construction are no longer real numbers at this value of n, which was explained in my previous post on the subject.

jestingrabbit: That's an example of an invalid limiting procedure of a shape. If a shape with a differentiable boundary is being successively approximated by some converging sequence of shapes, then the derivative of the boundary of approximations must also converge to the derivative of the resulting shape in the limit. You are constructing the diamond with a diagonal edge using increasingly fine squares with horizontal/vertical edges. It fails for the same reason that this does.

And again, the justification for that might be best thought of in terms of the limiting size of your squares. If you explicitly define the sequence of square approximations to the diamond, and fine the iteration at which the corners of your squares completely cover the sqrt(2) edge length, you will find that the squares you are adding at that iteration are a fraction of a single point in quantity. Such an object is disallowed.

At this point, quantity is undefined for unbounded sets like the natural numbers. I've put some effort into trying to make that work, but it brings it's own challenges. For example, if N denotes the quantity of natural numbers, then by using an argument from translation invariance, N+1=N. So it can't be a standard hyperreal number, and things get much more difficult.

But can't you just use that same argument from translation invariance on some subset of [0,1) that is equivalent to the natural numbers, like the subset {1/n | n in Z+} like benneh suggested? If you're not happy to keep moving points from infinity, why not move points towards infinity - so move the point at 1/2 to 1/3, and the point at 1/3 to 1/4, etc... ?

Unless the concept of 'quantity' you are trying to define depends on having some ordering or distance function defined on the elements of the set?

My previous comment didn't rely on the fact that the sequence (1/n) converged to 0. It could have been any sequence whatsoever. (1/n) was just convenient.Instead of using the sequence ([0,1] - {1/n}), you could use the sequence ([0,1] - {a_n}), where the sequence (a_n) isn't convergent.

I suppose you could build the natural numbers by pulling discrete points out of the unit interval, but that wouldn't give you any information about the quantity of natural numbers unless you could define the quantity of points you removed in order to construct them. And if you can't define the quantity of points removed, you can't define the quantity of points remaining in the interval, so it wouldn't be surprising for such a set to stop obeying the usual rules of quantity.

Edit: Or are you basically just restating the same thing benneh said about the quantity becoming x instead of x-1 after the infinite sequence of translations? If so, I've already addressed that. Any sequence in which you can make the 'gap' disappear can be reinterpreted as having the 'gap' converge to a non-real quantity, which is not guaranteed to leave quantity unchanged.

If you can 'reinterpret' sets like that, can't you just reinterpret the line [0,1) as an unbounded sequence of real numbers? It really does seem like the distance relation defined on the sets is integral to your definition of quantity.

arbiteroftruth wrote:I suppose you could build the natural numbers by pulling discrete points out of the unit interval, but that wouldn't give you any information about the quantity of natural numbers unless you could define the quantity of points you removed in order to construct them. And if you can't define the quantity of points removed, you can't define the quantity of points remaining in the interval, so it wouldn't be surprising for such a set to stop obeying the usual rules of quantity.

Surely it should give you the result that the quantity of the natural numbers is less than the quantity of the real numbers? Or do you not want the subset relation to preserve quantity?

When I've played around with extending quantity to unbounded sets, there are some other restrictions that apply. I figure quantity on the natural numbers should resemble natural density in many respects, and given that notion, distance is definitely important. It can basically be summed up by saying that quantity is only necssarily preserved under finitely many piecewise translations. Otherwise you could equate the natural numbers with any countable set, and you just have cardinality. So, if the natural numbers can be quantified at all, putting them in bijection with a subset of an interval wouldn't be a valid argument for saying that N is smaller than R1.

But that limitation on countably many piecewise translations shouldn't apply to bounded sets.

The "reinterpretation" comment was badly phrased. And probably the easiest way to address that issue is to think of non-included points being just as important to define as included points. So that with [0,1), if you move the point at 1/2 to 1, you've also moved the gap at 1 to 1/2. Thinking in those terms, the argument that after infinitely many steps there is no gap because any value the gap could take on has been filled, implies that the gap was moved out of the set of real numbers. And that's no longer a simple translation.

Informally, an infinite process that uses calculus-like logic of approaching a limit will preserve quantity, but an infinite process that uses set-theory-like logic along the lines of removing a feature because it never takes on a stable position will not preserve quantity.

Using the [0,1]-1/n example, the argument from set-theory-like logic is that there is no final value for 1/n, so after infinitely many steps we cannot identify any particular point to be removed from the interval. But the argument from calculus-like logic is that the value for 1/n converges to 0, so the limit of the sequence of intervals is (0,1], and quantity is preserved.

Perhaps the more rigorous way to state that is that the distance relation is important, because the limit of a sequence of sets in this context is to be taken as the set resulting from the limit of the positions of its various features, rather than the set-theoretic limit of the sequence of sets.

I think the question to ask isn't "can this be made rigorous?" but instead "once this is rigorous, what subsets does it apply to, and what properties does it have?" It seems that so far, the properties we want are (1) invariance under isometries and (2) finite additivity of disjoint sets. It looks like this is only going to work for bounded sets. To start with let's look at sets of the form (infinitely differentiable function) > 0, together with finite unions, intersections, and complements. Countable disjoint additivity may also work, but infinite disjoint additivity obviously doesn't, since [0,1] and [0,2] are both just the union of the same number of points.

I'm sure the 1-dimensional case works, and the reason is because we can give an alternative definition: given a closed set A of the above form, let q(A) = (total length of A)x + (number of components of A). We can then extend this to non-closed sets by subtracting points. Similarly, given a closed 2-dimensional shape B, we let q(B) = (total area of B)x2 + 1/2(length of the boundary of B)x + (euler characteristic of B).

Another interesting property you get is scale invariance: if f is a similarity of space that scales by t, then it seems like the correct relation is q(f(A))(x) = q(A)(tx) (thinking of q(A) as a function of the formal variable x). This should be enough to calculate q for fractals defined by self-similarity, such as the Cantor set.

Elmach wrote:Can you post how you derived the measure of the Cantor set, because as far as I can tell, it is

1+x - (1/3) sum(2n (1/3nx - 1))

This follows from countable disjoint additivity, it's just the unit interval minus the infinite disjoint union of open intervals. The x's cancel out in the sum, and you're left with 1 + sum(2n). This can't be right, because it implies that the quantity of the Cantor set doesn't depend on the length of the interval you start with. It also implies that the Cantor set has the same quantity as a countable set (the endpoints of the intervals of the finite approximations to the Cantor set). Without countable additivity, it's going to be very hard to calculate the quantity of interesting shapes. Maybe you can recover countable additivity if you throw out fractals and only look at nice shapes...but as the diamond example shows it's only going to work under a restricted case where you have some kind of C1 convergence of the boundaries.

Another issue that strikes me is the 3D case: I don't know how to define the quantity of even nice shapes. If I have a nice closed domain A, we want q(A) = (total volume of A)x3 + 1/2(surface area of the boundary of A)x2 + (???)x + (euler characteristic of A). I don't know what the (???) should be, but from the case of the cube we see that it isn't necessarily zero.

Edit: Here's a better way to view this Cantor set issue. Let C be the "open Cantor set", ie you start with the open interval (0,1) and you repeatedly delete the closed middle thirds. C is non-empty because 1/4 is in C (also every irrational number in the standard Cantor set is also in C). But if we assume countable additivity,

I think countable additivity works for quantities with no finite term. That is, you can sum [0,1)+[2,2.5)+[3,3.25)+... to get q=2x, but you cannot do the same with [0,1]+[2,2.5]+[3,3.25]+..., because each closed interval is of the form r*x+1. More generally, if any term in the summation diverges to infinity, then countable additivity fails. It can be recovered if we require such cases to associate the specific countable infinity of the sequence with a particular infinite hyperreal in terms of x.

In the Cantor set example, we can't simply cite countable additivity when subtracting open intervals, because each open interval is of the form r*x-1. The summation "converges" to x-(divergent summation of endpoints). The quantity of the Cantor set is exactly the quantity of that divergent summation, whatever that is. And that can only be defined if we provide some justification for the countable infinity to take on a specific hyperreal value in that scenario.If we extend the formula that q([a,b))=(a-b)*x to work for hyperreal values of a and b, and argue that the construction of the Cantor set ends when the quantities of the removed open intervals are 0, then that occurs when the intervals are of the form (r,r+1/x). By the formula above, q([r,r+1/x))=1, so the equivalent open interval has quantity 0. Then we can calculate the exact value of n that would lead to such 0-quantity intervals, set that as the countable infinity for this particular construction, and compute the quantity of the Cantor set.

Perhaps I should restate my post, I haven't seen any response to it yet: you do not have additivity at all, as you have stated the quantity function.

My counterexample is simple. Let [imath]q(S)[/imath] be the quantity of a set. Then we know that [imath][0,2) = [0,1)\cup[1,2)[/imath]. Clearly [imath]q([0,2)] = 2R_1+1[/imath] but [imath]q([0,1)) + q([1,2)) = 2R_1[/imath].

I do not think this can be fixed: in fact, I believe there is a trade-off between things like additivity and making sense of point-wise measures.

Finally, I want to know how you propose to consider the set of all real numbers in [0,1), with all the rational numbers removed. Is the quantity 0 or R_1? Or perhaps something in between? Or is it "unquantifiable"? It seems that the "definition" of quantity is so muddled up as is that it needs constant clarifications or special cases from you... I find the concept interesting, especially if we could rid ourselves of such lamentable plagues as unmeasurable sets, but it seems that it cannot be strictly defined as is.

I get 2R_1. It doesn't have "scalability" or whatever it would be where Cq(S) = q({Cx | x\in S}), which maybe makes the right question "what is q([0, 1/2))?"(dunno what I was thinking there, it does have scalability for half open sets).

--

I think that there is some sense in the cantor set construction here, but I don't get the "discard the coefficient" step.

Also, talking about the hyperreals is counterproductive. I don't see any evidence of R_1 being in there, I mean, what is its equivalence class in the hyperproduct construction? You could maybe claim that its in the equivalence class of (1, a, a^2, a^3,...) for some a, but which a? I would just claim that R_1 is some symbol that behaves something like a real number and leave it at that for the moment.

Yeah I wouldn't chose a hyper real number. I don't know much about hyper reals but the ultraproduct construction (and therefore the field of resulting hyperreals IIRC) depends on the choice of an ultrafilter. It is more convenient to work in the polynomial ring R[x] and treat the variable x as R1.

I'm not convinced your notion of "taking limits of quantity" is a good choice. I don't see how it is possible to calculate the quantity of S1 for example. Maybe a better condition would be: Quantity is preserved by differentiable functions f: U -> Rm with |f'(u)| = 1 on U. But then again being differentiable is a relatively strong condition. It would be easier to replace that by a definition that only depends on the topology of R.Also note that you didn't prove that quantity is well defined at all. In order to prove that you need to show that there is only one function q from of subset of 2R to R[x] that satisfies q([0, 1)) = 1 and your invariance conditions.

Let's consider the following intervals In=[0,1/n). The quantity of such an interval should be q(In)=1/n R1. Now, in the limit(intersection over all n) In is [0,0] or the single point 0, which has a quantity of 1, whilst the limit of the quantities is 0, assuming limits are linear over R1.

I think you're better off using open intervals to define R1 and then postulate that only the open sets in any set are measurable, this however, severely constrains the measurable sets. Still I would suggest that you formalise which axioms you want your quatity to satisfy.

Also there is a famous theorem that there can't be a measure on the real numbers having the (compactified) positive real numbers as codomain. So R1 can't be real (no pun intended)

Let x be a positive infinite hypernatural number with the property that r*x is also a positive infinite hypernatural number for any positive real number r.

Define a function q from certain subsets of Rn to *N, where the value q(S) for an appropriate set S is called the "quantity of S".

Define q( [0] ) = 1, and q( [0,1) ) = x

Define q(S')=q(S) where S' is a combination of translations, rotations, and reflections of S within Rn. (invariances)

Define q(S1+S2)=q(S1)+q(S2), where S1+S2 is the disjoint union of sets S1 and S2. (finite additivity)

For sets S1 in Rn, and S2 in Rm, define q(S1 x S2)=q(S1)*q(S2), where S1 x S2 is the cartesian product in Rn+m. (extension from [0,1) to higher dimensions)

An infinite sum of sets is said to be quantitatively convergent if, when the quantities of the individual sets are expressed as polynomials of x, the infinite sum of each coefficient in the polynomial is convergent. Define the quantity of a quantitatively convergent infinite sum of sets as the limit of partial sums. (restricted countable additivity)

An infinite sequence of sets is said to be quantitatively convergent if the following conditions hold:1. The quantities of the sets converge to a particular polynomial of x.2. The quantity of the symmetric difference of the nth set and the n+1th set converges to 0 as n tends to infinity(for constant m, the quantity of r*xm is said to converge to 0 if r converges to 0, even though for every positive r, r*xm is infinite).

If a sequence of sets is quantitatively convergent, define the quantitative limit of the sequence to be the set containing every point p for which:1. Given any positive r, after a sufficient number of sets in the sequence, all subsequent sets in the sequence include points within the sphere of radius r centered at p, and for a sufficiently small r, after a sufficient number of sets in the sequence, no subsequent set has any non-included boundary within the sphere of radius r centered at p, OR2. The quantitative limit of the sequence of the sets of included boundary points contains p.

Define the quantity of a quantitative limit of a sequence of sets to be the limit of the quantities of the sets in the sequence.

These rules for quantity, assuming they result in a consistent and well-defined quantity function, should allow quantity to be defined for any "nice" bounded shape. For now, quantities of fractals, unbounded sets, and other less-nice sets are undefined because they will not meet the requirements of quantitative convergence. I still think the quantities of fractals can be defined by meaningfully assigning a specific infinite hyperinteger value (in terms of x) to the number of steps in the construction, but I'll have to think more about how to define the method of assigning such a value, so for now let's leave them undefined.

The criteria for quantitative convergence of sequences of sets and the definition of the quantitative limit set should address most of the issues that have been brought up so far. The [0,1/n)U(1/n,1] example fails to converge because the quantity of the difference between consecutive sets in the sequence is always 1. Filling a diamond with squares does not converge because the coefficient of x1 does not converge. For now, the Cantor set and other fractals are left undefined because some coefficient fails to converge. But the sequence of regular n-gon approximations to the circle does converge to the circle, and the sequence of filled n-gon approximations to the disk converges to the disk.

EDIT:

I think I've worked out how to extend this to fractals.

An infinite sum of sets that is not quantitatively convergent is said to be pseudo-convergent if the following conditions hold:1. After a sufficient number of sets in the sum, all subsequent sets are finite unions of disjoint sets of equal quantity q, where q is expressible as a function of n and converges to 0 as n tends to infinity.2. The function of n for the value of q can be solved in the hyperreals for q=0.3. For sufficiently large n, the quantity of the sum of the first n sets in the summation can be expressed as a function of n.Define the quantity of a pseudo-convergent sum of sets to be the polynomial of x in which each coefficient of x is equal to the limit of partial sums of that coefficient if it converges, and is otherwise equal to the value calculated by using the formula for partial sums with n such that q=0.

Given this definition, the Cantor set can be quantified as follows.

Construct the Cantor set by starting with [0,1] and subtracting middle thirds. The quantity of the nth term is x+1-sum(2n(3-n-1x-1)), where the nth set of middle thirds consists entirely of disjoint open intervals of quantity 3-n-1x-1. Setting this quantity equal to 0 and solving for n, we get n=log3(x)-1. Evaluating the summation by ordinary methods for any coefficient where the sum converges, what remains is 1+sum(2n). Using n=log3(x)-1 and the formula for partial sums of a geometric series, the final quantity can be calculated as xlog3(2).

Let's forget about infinite sequences of sets for a minute. Let's just consider a plain unit circle (not filled in) in R^2, call it C.

Consider a point x on C, and consider the function f that rotates the circle by, say, sqrt(2) degrees. Then if you keep applying f to x, you never get back to x itself.

Define the set A to be all the points you can get to by starting at x and applying f zero or more times. Now write C = A union C-A. Note that f(A) = A - {x}.

Correct me if any of the following steps are wrong under your definitions:

1) Since quantity should be invariant under rotation, this means the quantity of A is equal to the quantity of A - {x}.2) Since quantity should be additive for two disjoint sets, the quantity of the unit circle C is equal to the sum of the two quantities of A and C - A.3) Similarly to 2), the quantity of C - {x} is equal to the sum of the two quantities of A- {x} and C - A.4) So by the previous steps, we get that the quantity of C is equal to the quantity of C - {x}.

So the circle has the same quantity as the circle minus a point?

I can keep working this towards the Banach-Tarski paradox if we like.

Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

Those steps are valid if the quantity of A is defined. I don't think it's possible to assign a quantity to A given the rules I've outlined so far. So I take your post as a proof that q(A) is undefined.

I expected that Banach-Tarski would still be possible. And the resolution is the same as usual: define quantity in such a way that any set that leads to the paradox is unquantifiable, just as measure is defined in such a way that sets leading to the paradox are unmeasurable.

I still don't think you can avoid the Cantor set problem with anything you've mentioned so far: Let C be the "half-open Cantor set". That is, we start with [0,1), then remove [1/3, 2/3), and on and on. C is an uncountable set. But q(C) = x - 1/3 sum(x(2/3)n) = 0.

I believe that if you follow the rules for quantitative convergence, you will find that that construction converges to the empty set. The set you get from the set-theoretic construction is not necessarily the same as the set you get in quantity theory using the same construction.

EDIT: Specifically, consider the point 2/3. This is not included in any of the half-open intervals you subtract out. But endpoints arbitrarily close to it are removed. You get [7/9,8/9), then [19/27,20/27), then [55/81,56/81), and so on. The left endpoints of these intervals converge toward 2/3, so the infinite sum of middle thirds is defined to include it as well.

Granted, that argument does require extending the ideas of quantitative convergence to non-convergent sets, but without extending those ideas in some way, there wouldn't be *any* definition of what set you get in quantity theory when using a fractal construction, so there still wouldn't be a paradox.

MartianInvader wrote:Let's forget about infinite sequences of sets for a minute. Let's just consider a plain unit circle (not filled in) in R^2, call it C.

Consider a point x on C, and consider the function f that rotates the circle by, say, sqrt(2) degrees. Then if you keep applying f to x, you never get back to x itself.

Define the set A to be all the points you can get to by starting at x and applying f zero or more times. Now write C = A union C-A. Note that f(A) = A - {x}.

Correct me if any of the following steps are wrong under your definitions:

1) Since quantity should be invariant under rotation, this means the quantity of A is equal to the quantity of A - {x}.2) Since quantity should be additive for two disjoint sets, the quantity of the unit circle C is equal to the sum of the two quantities of A and C - A.3) Similarly to 2), the quantity of C - {x} is equal to the sum of the two quantities of A- {x} and C - A.4) So by the previous steps, we get that the quantity of C is equal to the quantity of C - {x}.

So the circle has the same quantity as the circle minus a point?

I can keep working this towards the Banach-Tarski paradox if we like.

Yeah, we've gotta get rid of these monsters. Lot of things don't work...

TwistedBraid wrote:I still don't think you can avoid the Cantor set problem with anything you've mentioned so far: Let C be the "half-open Cantor set". That is, we start with [0,1), then remove [1/3, 2/3), and on and on. C is an uncountable set. But q(C) = x - 1/3 sum(x(2/3)n) = 0.

Or we could just say that it doesn't satisfy countable additivity/limits.

Edit: its the dominated convergence that its not satisfying. And that's final.

I need to refine the definitions of quantitative convergence of sets. Let's see how the following version works.

The limit set to which a sequence of sets quantitatively converges, if the sequence converges, is the set containing every point p for which:1. Given any positive r, after a sufficient number of sets in the sequence, all subsequent sets in the sequence include points within the sphere of radius r centered at p, and for a sufficiently small r, after a sufficient number of sets in the sequence, no subsequent set has any non-included boundary within the sphere of radius r centered at p, OR2. The limit set of the sequence of the sets of included boundary points contains p.

But the sequence of sets only quantitatively converges if it meets additional conditions.

Let the normal set of a set S in Rn be the set of all unit vectors normal to the boundary of S (whether those boundary points are in S or not), mapped to points in R2n. The first n coordinates being the location of the boundary point, and the second n coordinates being the magnitudes of the unit vector normal to that point along each dimension.

A sequence of sets converges to its limit set if and only if:1. The limit set is well defined as described above, and2. The sequence of normal sets of the original sequence quantitatively converges to the normal set of the limit set.

This new condition covers cases like filling a diamond with squares or approximating the circle with right angles. Incidentally, it also covers the case of [0,1/n)U(1/n,1], because the normal sets of this sequence always contain four elements, while the normal set of the limit set [0,1] only has two elements.

Given that, I think I can remove one of the requirements I listed in an earlier post. I had said that one requirement was that the quantity of the symmetric difference of the nth set and the n+1th set converges to 0 as n tends to infinity. I don't think that's necessary anymore.

The method of computing quantity for pseudo-convergent sequences remains the same, but the rules for determining the limit set are the same for pseudo-convergent sequences as they are for convergent sequences.

Given these requirements and definitions of convergence in quantity theory, I think all the problem cases that have been put forward can be addressed.

MartianInvader's set A cannot be assigned a quantity using the existing rules for assigning quantity, and I believe this is true for other Banach-Tarski-esque sets as well.The regular Cantor set can be quantified as xlog32 using the rules for quantifying pseudo-convergent sets.The construction of the half-open Cantor set fails to converge, because the limit set is the empty set, but the limit set of the sequence of normal sets has infinitely many elements.Filling a diamond with squares, or approximating a circle with right angles, both fail to converge because the limit set of the sequence of normal sets does not match the normal set of the limit set.[0,1/n)U(1/n,0] fails to converge for the same reason. The limit set is [0,1], with a normal set of 2 elements. But the limit set of normal sets of the sequence has 4 elements.

But the definitions do still allow quantity to be defined for all sorts of "nice" shapes.Approximations to the circle or disc using regular n-gons converge and give an intuitive quantity of the circle or disc.Approximations to the cylinder using regular n-gonal prisms converge and give the same quantity as constructing the cylinder as the product of a disc and a closed interval.Cones can be constructed and quantified as the limit of pyramidal approximations.

For 2 dimensional shapes, its possible to implement a form of integration using quadrilateral approximations of progressively finer intervals. It is likely possible to extend this to higher dimensions, although it starts to hurt my head.

That's a good point. You could still construct it, but without an actual interior part of the set, there are no normal vectors to distinguish an n-gonal approximation from a right-angle approximation. The notion of the normal set could be adjusted to look at the vectors normal to the set itself, rather than just its boundary, but then the dimensionality of the normal set is the same as that of the original set, and there's an infinite regression of sets to check for convergence.

It's probably better to go with convergence of the derivative in some way. Informally, the directions you can move from a given point without leaving the set should converge to the same directions you can move in a limit set, but the criteria for convergence of those sets of vectors do not include the same requirement.

For a given point p in a set S, let Vp be the set of unit vectors oriented such that for a sufficiently small radius r, the point r distance from p along the vector is still part of the set S. Let V be the set of all sets Vp for all p in S.

A sequence of sets converges to S if it meets the previously described criteria for including the same points as S, and if the sequence of vector sets meets the same criteria for including the same vectors as V, when such sets of vectors are mapped to sets of points in higher dimensional space.

There may still be some need for refinement, but surely there is a way to define the criteria for convergence such that the "nice" constructions work without running into contradictions when attempting the more malicious constructions.

EDIT:

Okay, for real this time, I think I've got it figured out.

Define x as a positive infinite hypernatural such that r*x is also a positive infinite hypernatural for any positive real number r.Define a function q from certain subsets of Rn to *N with properties described below, where q(S) can be called the quantity of S.q([0])=1, and q([0,1))=xq(S')=q(S) where S' is a combination of translations, rotations, and reflections of S.q(S1+S2)=q(S1)+q(S2), where S1+S2 is the disjoint union of S1 and S2.q(S1xS2)=q(S1)*q(S2), where S1xS2 is the cartesian product of S1 and S2.

Define the quantitative limit set of a sequence of sets to be the set containing all points p according to the following rules.1. If, after a sufficient number of sets in the sequence, all subsequent set contain p, then the limit set contains p unless p is excluded by other criteria described below.2. Consider a sphere of arbitrarily small radius r and centered at p. If, after a sufficient number of sets in the sequence, all subsequent sets have a non-included boundary point within the sphere, and no subsequent sets have an included boundary point within the sphere, then p is not included in the limit set.3. If, after a sufficient number of sets in the sequence, all subsequent sets have an included boundary point within the sphere, and no subsequent sets have a non-included boundary point within the sphere, then p is included in the limit set.4. If, after a sufficient number of sets in the sequence, all subsequent sets have non-included and included boundary points within the sphere, and there is a path between them that leaves neither the sphere nor the set, then p is not included.5. If, after a sufficient number of sets in the sequence, all subsequent sets have non-included and included boundary points within the sphere, and there does not exist any path between them that leaves neither the sphere nor the set, then p is included.6. Regardless of the criteria above, if p is in the quantitative limit set of the sequence of sets of included boundary points of sets of included boundary points(repetition intentional) of the original sequence, then p is included.If, given a sphere of arbitrarily small radius r and centered at p, the sequence has infinitely many sets that include points in the sphere and infinitely many sets that don't include points in the sphere, the the quantitative limit set of the sequence does not exist. Likewise, if there are infinitely many sets with non-included boundaries within the sphere, and infinitely many without non-included boundaries within the sphere, then the quantitative limit set does not exist.

Define the vector set of a set S in Rn as follows.For a point p in S, consider a sphere of radius r centered at p. Consider the intersection of S and the sphere. Next, consider the set of unit vectors positioned at p and oriented towards the points of intersection between S and the sphere. Map this set of unit vectors to R2n by setting the first n coordinates equal to the coordinates of p, and setting the next n coordinates equal to the components of the unit vectors. Take the union of such sets of mapped unit vectors for all p in S. This is the vector set of S.

A sequence of sets is said to quantitatively converge if the quantitative limit set exists and if the vector set of the quantitative limit set is the same as the quantitative limit set of the sequence of vector sets.

A sequence of quantities is said to converge if it converges to particular expression in terms of x.

Given a set S that is the quantitative limit set of a quantitatively convergent sequence, define q(S) to be the limit of quantities of the sequence, if the sequence of quantities converges.

A sequence of sets is said to be pseudo-convergent if the following properties hold:1. The sequence is quantitatively convergent, but the sequence of quantities does not converge.2. The sequence can be expressed as the sequence of partial sums of an infinite sum of sets.3. For sufficiently large n, the nth set in the sum and all subsequent sets consist of disjoint unions of a finite number of sets of equal quantity qn.4. qn can be expressed as a function of n and solved in *R for qn=0.

Given a set S that is the quantitative limit set of a pseudo-convergent sequence, define q(S) as follows.Using n to specify the number of steps taken through the sequence, take the limit of the sequence of quantities as an expression of both x and n as n tends towards infinity. Then, find the value of n such that qn=0, and substitute this value into the expression for the quantity. The resulting expression in terms of x is q(S).

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Given these definitions, "nice" shapes can be quantified using a sequence of "nice" approximations, but a sequence of jagged approximations will not converge, because the vector sets won't match. This covers cases such as right-angle approximations to the circle or filling a diamond with squares.

The sequence defined by [0,1/n)U(1/n,1] converges to (0,1], so the quantity comes out right. The same occurs for any such sequence where the gap converges to a particular value. If the gap moves around in some malicious way, such as by moving through points in the Cantor set in some lexicographic order, then the sequence will fail to converge, because the Cantor set is in some sense dense at individual points, even though it is not dense on any interval. That is, taking the value 2/3 as an example, the Cantor set contains infinitely many points arbitrarily close to 2/3, so the "gap" will neither permanently converge to 2/3 nor permanently "stay away". So such a sequence of sets fails to converge.

The regular cantor set can be quantified by using pseudo-convergence of the infinite sum of middle thirds, then subtracting this from [0,1]. The construction of the half-open Cantor set ends up converging to the empty set. The sum of half-open middle thirds converges to [0,1], so after subtracting it from [0,1] the result is the empty set.

The set of points of angle n*sqrt(2) around the circle cannot be quantified. Any attempt to construct it or a similar set will quantitatively converge to the entire circle. So such a set cannot be used to create a Banach-Tarski-esque problem.

The rules for which points are in the limit set are a bit complex, but they are motivated by fairly simple intuitions. Essentially, if a non-included boundary converges to a point, that point should not be included, and if an included boundary point converges to a point, that point should be included. If both types of boundaries converge to the same point because the sequence keeps shrinking down the set until there is no interior separating the boundaries, then the point should not be included, because the set has been shrunk out of existence. If both types of boundaries converge to the same point because the sequence keeps shrinking a gap between two parts of the set until there is no gap separating the boundaries, then the point should be included, because the gap has been shrunk out of existence. Finally, if both types of boundaries converge to the same point because that point is at an interface between included and non-included portions of the overall boundary, then the inclusion of the point depends on an analysis of that interface as its own sequence of sets. And if the boundary points don't converge to particular values at all, the sequence should not converge to any particular set.

This means that [0,1/n) converges to the empty set, [0,1)U[1+1/n,2+1/n) converges to [0,2), a sequence of open regular n-gons converges to an open disc, a sequence of closed regular n-gons converges to a closed disc, and a sequence of half-open regular n-gons converges to a half-open disc, with the endpoints of the included part of the circumference being either included or not depending on whether they were included in the n-gons. All of these types of convergences seem intuitive to me, so it was important to define convergence in a way that agrees with all of them.